Effective Conductivity of a Cloudy Ionosphere - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 300

10.3 Inhomogeneous Plasma

Stochastic Inhomogeneities

Let us consider a stationary current in a medium in which the conductivity
tensor σ ( r ) is a random function of the coordinates. We will restrict our
consideration to analysis of small-scale disturbances with length-scale much
smaller than the size of the system. Then Ohm's law connecting local current
density j ( r ) and local electric field E ( r ) can be written as

j ( r )= σ ( r ) E ( r ) . (10.12)

In addition, j ( r )and E ( r ) satisfy the equations

∇ · j ( r )=0 , ∇ × E ( r )=0 , E = − ∇ ϕ, (10.13)

where ϕ is a potential.
Much more interesting, however, is not the relation between specific lo-
cal conductivity and local electric field (10.12) but rather the relation (10.4)
between average current j i and average electric field E k :

= σ eff
ik

j i E k . (10.14)

Here σ eff

ik is the effective conductivity of a spatially inhomogeneous anisotropic
system. We define the fluctuation δy ( r ) and the mean-free value y of y ( r )

as such that

+ δy ( r ) , (10.15)
where y ( r ) is any variable. For example, the potential ϕ can be written as

y ( r )= y

ϕ = ϕ + δϕ = − E k x k + δϕ, (10.16)

then

∂ϕ
∂x k

+ ∂δϕ = − E k

∂x k .

(10.17)

Assuming that

σ ik = σ ik + δσ ik (10.18)

we have for j i :

+ δσ ik ) σ ik ∂ϕ

∂x k

+ ∂δϕ
∂x k

j i = − = − ( σ ik − E k

∂δϕ
∂x k

δσ ik ∂δϕ = σ ik E k − σ ik + δσ ik E k −

∂x k .

j i

Then an expression for the mean current can be written as

∂δϕ
∂x k

+ δσ ik ∂δϕ

∂x k

.

j i = σ ik E k − σ ik δσ ik E k − (10.19)